This page explains the mathematics behind the quadraphonic matrix systems. The mathematic principles used are based on algebra, complex numbers, and trigonometry.
Calculating decibels:
The formulas in this section are in Microsoft Excel ^{®} form.
Value_dB = 20 * log10(voltage_ratio)
Procedure: Find the common logarithm of voltage_ratio. Then multiply by 20.
Value_dB = 10 * log10(power_ratio)
Procedure: Find the common logarithm of power_ratio. Then multiply by 10.
voltage_ratio = 10 ^ (Value_dB /20)
Procedure: Divide Value_dB by 20. Then raise 10 to the power of the result.
power_ratio = 10 ^ (Value_dB /10)
Procedure: Divide Value_dB by 10. Then raise 10 to the power of the result.
In quadraphonics, this would be 4 equations in 4 unknowns  a discrete quadraphonic system.
Such a system can have no solution. But such systems are not used in quadraphonics.
In quadraphonics, this would be 2 equations in 4 unknowns.
In quadraphonics, this would be 4 equations in 2 unknowns.
Modulations on the stereo record groove are defined by the RIAA (Recording Industry Association of America).
Here are the stereophonic groove modulations as defined by the RIAA:
These modulations are as viewed from beyond the cartridge end of a normally mounted pickup arm:
The left channel is recorded on the side of the groove closest to the
spindle.
 Left channel modulations slope up to the right and down to the left at a
45° angle (cyan).
The right channel is recorded on the side of the groove closest to the
rim.
 Right channel modulations slope up to the left and down to the right at a
45° angle (red).
Inphase modulations are lateral (horizontal  olive). They emulate a monaural record.
Reversed phase modulations are vertical (violet).
Clockwise motion (viewing the cartridge end of the tonearm) occurs when the left channel leads the right channel by 90° (black).
Anticlockwise motion (viewing the cartridge end of the tonearm) occurs when the left channel lags the right channel by 90° (brown).
 Special equipment (including a special pickup cartridge) must not be needed to play the recording in stereo.
 Very few people will buy a recording not compatible with normal stereo equipment.
Only quadraphonic enthusiasts would be able to use it. So it is essential to make recordings fully playable on stereo players.
These rules ensure compatibility:
 This is necessary to keep from overloading one side of the stereo recording.
 The encoding and decoding methods should accomplish this.
 This is so the two stereo speakers will aid each other in reproducing the bass.
 At least two of the quadraphonic speakers should be in phase to aid each other in reproducing the bass.
 In addition, if the recording is on a phonograph record, the bass must be recorded laterally (in phase) in the groove.
Bass recorded in other orientations must be recorded at much lower levels to avoid overcutting the groove.
 This is a problem with most encoding systems.
 Usually the recording engineer is advised to not pan important material to center back.
Center back is a good place to hide some extra reverb.
 Large phase differences make a weird "phasiness" sensation when playing the recording in stereo.
 This effect makes image location of musical parts difficult.
 This unwanted effect happens with the UMX (BMX), Matrix H, Matrix E, Matrix G, and Phase Location.
 Note that while the matrix decoder removes phase anomalies when decoding the record, the stereo player does not.
 It is not advisable to use a matrix encoding system that shifts the phase of the front channels.
 A radiofrequency carrier in the groove causes a swooping sound when a phonograph record is staticcued or slipcued.
Radiofrequency carriers should not be used.
Nothing in the encoding and decoding system may change the basic waveforms that are encoded. This means that the following mathematical operations must not be used on the signals:
* Except where the trigonometric value is a constant used in a polynomial expression.
The process used for quadraphonic matrix is called matrix multiplication.
The following are the matrices used for one encode and one decode operation in matrix quadraphonics:
Here is an example of this use of the matrix. The ElectroVoice Stereo4 System is shown. But it works for any matrix system.
ENCODER  DECODER  


The procedure used to calculate this is found at the matrix multiplication page.
The rules of complex mathematics;





The Poincaré Sphere (also called the Stokes Sphere, the Foucault Sphere, and the Fresnel Sphere) was originally conceived by Henri Poincaré ("onREE pwancarRAY") in 1892 to describe polarized light, and by Foucault ("fooCO") to describe freeswinging pendulum motion. Stokes and Fresnel ("frehNELL") independently discovered its use for polarized light. Scheiber and Robinson independently discovered it for use in matrix quadraphonic signals. The quadraphonic stylus motions are analogous to the Foucault pendulum motions.
The Poincaré Sphere, representing the two channels of a stereo recording or phono stylus modulations, represents stylus motions on a phonograph record and signal relationships in other media. The diagrams at the right are as follows:
These are the descriptions of the six orthogonal motions and their positions on the Poincaré Sphere:
Any plane passing through the center of the Poincaré Sphere produces a great circle where it intersects the Poincaré Sphere.
The following matrix systems are examples of GreatCircle Matrix systems. The order of the listed colors in the table shows the movement on this Poincaré sphere (shown at right) of a sound panned clockwise around the listener starting at the front. Large spots are on the far side of the sphere.
MATRIX  ORIENTATION  COLORS ON DIAGRAM AT RIGHT  

FRONT  RIGHT  BACK  LEFT  FRONT  
QS, Stereo4, Dynaquad  Equator  olive  red  violet  cyan  olive 
UMX and BMX  Opposite Meridians  black  red  brown  cyan  black 
BBC Matrix H (see below)  45° diagonal  orange  red  blue  cyan  orange 
Matrix HR (see below)  45° opposite diagonal  pink  red  yellow  cyan  pink 
Phase Location (Denon experiment)  Opposite Meridians  olive  brown  violet  black  olive 
Most other matrix systems are not greatcircle systems.
The relative phase angles are as follows:
The following formula calculates the geodesic distance between two points on the surface of the sphere. This geodesic distance gives a central angle, which is used in the formulas below:
The variables:
The calculations:
Examples of orthogonal pairs (see diagram above):
Proof for the following three orthogonal pairs:
These are the original left and right encoded channels, recorded in the left wall and the right of the stereo record groove using the Westrex 45/45 process. Since no signal processing was done before making the recording, they are still the original unchanged L and R signals from the quadraphonic encoder.
L' = L  R' = R 
The created orthogonal signals are signals from a sum and difference matrix:  
H = 0.71 L + 0.71 R  V = 0.71 L − 0.71 R 
This orthogonal pair is used for FM multiplex stereo.  
The original signals are recovered using a sum and difference matrix:  
L' = 0.71 H + 0.71 V  R' = 0.71 H − 0.71 V 
L' = 0.71 (0.71 L + 0.71 R) + 0.71 (0.71 L − 0.71 R)  R' = 0.71 (0.71 L + 0.71 R) − 0.71 (0.71 L − 0.71 R) 
L' = 0.5 L + 0.5 R + 0.5 L − 0.5 R  R' = 0.5 L + 0.5 R − 0.5 L + 0.5 R 
L' = L  R' = R 
The created orthogonal signals are signals from a quadrature matrix:  
A = 0.71 L + 0.71 Rj  B = 0.71 R + 0.71 Lj 
The original signals are also recovered using a quadrature matrix:  
L' = 0.71 A − 0.71 Bj  R' = 0.71 B − 0.71 Aj 
L' = 0.71 (0.71 L + 0.71 Rj) − 0.71j (0.71 R + 0.71 Lj)  R' = 0.71 (0.71 R + 0.71 Lj) − 0.71j (0.71 L + 0.71 Rj) 
L' = 0.5 L + 0.5 Rj + 0.5 L − 0.5 Rj  R' = 0.5 R + 0.5 Lj + 0.5 R − 0.5 Lj 
L' = L  R' = R 
Proof: Any monophonic signal taken from 2channel program material must be at some location on the Poincaré Sphere. Because of this, any material at the point orthogonal to the point used must completely disappear from the monophonic signal.
The usual way to create a mono signal from a stereo signal is to use this sum:
M = 0.71 L + 0.71 R
But the following signal is entirely absent from the mono signal:
0.71 L  0.71 R
Sansui used the sum of all of the decoder outputs, which reproduced the clockwise stylus motion and other motions that put sound into the clockwise reproduction
The anticlockwise stylus motion (not usually used) fully disappears from this signal.
The following are true of any two points on the Poincaré Sphere:
Example 1:
 Note that when ∠α is 0°, cos(α/2) = 1.
 Note that when ∠α is 180°, cos(α/2) = 0.
Example 2:
 Note that when ∠β is 0°, cos(β/2) = 1.
 Note that when ∠β is 180°, cos(β/2) = 0.
Here are some of the commonly used values:
Sphere Angle  0  30  39  45  54  60  82  90  109  120  133  135  143  150  151  160  169  176  179  180 

Crossfeed  1.00  0.97  0.94  0.92  0.89  0.87  0.75  0.71  0.58  0.50  0.40  0.38  0.32  0.26  0.25  0.18  0.10  0.03  0.01  0.00 
Separation dB  0.0  0.3  0.5  0.7  1.0   1.2   2.5  3.0  4.7  6.0  8.0  8.3  10.0  11.7  12.0  15.0  20.0  30.0  40.0  60.0 
Front separation:
Back separation:
Front to back separation:
Front separation:
Back separation:
Front separation:
Back separation:
Mono to back separation:
Back attenuation:
Any position attenuated more than 10 dB effectively disappears from mono playback.
Here are separations calculated using these methods.
Procedure:
C = cos(α/2)
Procedure:
S = 20 log10(C)
Procedure:
S = 20 log10(cos(α/2))
Procedure:
C = 10^(S/20)
Procedure:
α = 2 arccos(C)
Procedure:
α = 2 arccos(10^(S/20))
The colors in this list apply to only the diagram at right.
The angle θ (theta) is the angle from the L pole (cyan) of the sphere to the desired point C.
The angle θ is also the angle from the R pole (red) of the sphere to the corresponding desired point C'.
C' is 180°θ from the L pole.
The angle ø (phi) is the phase angle between L and R on the equator of the sphere, where:
The angle ρ (rho) is a phase shift added so the encoded signal has a wanted phase relationship to other signals.
The angles are converted into encoding sinecosine variable pairs:
The leftside left and right encoding outputs for point C, using signal x, is:
L = atx R = bftx
The corresponding mirrorimage rightside left and right encoding outputs for point C', using signal y, is:
L = bfty R = aty
Obtain the same angles used in the encoding equations above.
The angle θ (theta) is the angle from the L pole (cyan) of the sphere to the desired point C.
The angle θ is also the angle from the R pole (red) of the sphere to the corresponding desired point C'.
C' is 180°θ from the L pole.
The angle ø (phi) is the phase angle between L and R on the equator of the sphere, where:
The angle ρ (rho) is a phase shift added so the encoded signal has a wanted phase relationship to other signals.
The angles are converted into decoding sinecosine variable pairs:
The decoding output for point C, using signals L and R, is:
U = atL + bftR
The corresponding mirrorimage decoding output for point C', using signals L and R, is:
V = bftL + btR
In these equations, a, b, c, and d are values between 1 and 1.
In these equations, s and t are sign values of +1 or 1. Variable s goes with X, and t goes with Y.
In these equations, u and v are sign values of +1 or 1. Variable u goes with L, and v goes with R.
In these equations, m and n are sign values of +1 or 1. Variable m goes with Lj, and v goes with Rj.
For the encoding of any pair of symmetric signals, the following equations are used:
For the decoding of any pair of symmetric signals, the following equations are used:
Examples include the following matrix systems:
PAIR  a  b  c  d  s  t  u  v  m  n 

lf and rf  0.92  0.00  0.38  0.00  +1  +1  +1  +1  +1  +1 
lb and rb  0.00  0.92  0.00  0.38  +1  +1  +1  1  +1  +1 
Encoding equations:
Decoding equations:
PAIR  a  b  c  d  s  t  u  v  m  n 

lf and rf  1.00  0.00  0.00  0.00  +1  +1  +1  +1  +1  +1 
lb and rb  0.00  0.71  0.71  0.00  1  +1  +1  +1  +1  +1 
Encoding equations:
Decoding equations:
PAIR  a  b  c  d  s  t  u  v  m  n 

lf and rf  0.85  0.35  0.35  0.15  +1  +1  +1  +1  +1  1 
lb and rb  0.35  0.85  0.15  0.35  +1  +1  +1  +1  1  +1 
Encoding equations:
Decoding equations:
Note the following special values in the above matrices:
No matter which greatcircle matrix is used, there is at least one point where panning must be interrupted for a phase change as the sound is panned completely around the listener:
No matter which great circle is used, either the phase is continually changed as the signal is panned, or the phase must be changed at some point. If all of the phase changes are at zero crossings, then the final phase of the signal after the panning will be reversed from the original signals before the panning, as shown here:
DIR  F  RF  R  RB  B  LB  L  LF  F 

L  0.71  0.38  0.00  −0.38  −0.71  −0.92  −1.00  −0.92  −0.71 
R  0.71  0.92  1.00  0.92  0.71  0.38  0.00  −0.38  −0.71 
Examples:
The red bar shows the encoding phase hole.
DIR  F  RF  R  RB  B  B  LB  L  LF  F  

L  0.71  0.38  0.00  −0.38  −0.71  0.71  0.92  1.00  0.92  0.71  
R  0.71  0.92  1.00  0.92  0.71  −0.71  −0.38  0.00  0.38  0.71 
The orange bars are where the bus selector is changed when panning.
DIR  F  RF  R  R  RB  B  LB  L  L  LF  F  

L  0.71  0.38  0.00  −0.00  −0.38  −0.71  −0.92  −1.00  1.00  0.92  0.71  
R  0.71  0.92  1.00  1.00  0.92  0.71  0.38  0.00  0.00  0.38  0.71 
The orange bars show the encoding phase holes, automatically adjusted by phasors in the encode equations.
DIR  F  RF  R  R  RB  B  LB  L  L  LF  F  

L  0.71  0.38  0.00  0.00j  0.38j  0.71j  0.92j  1.00j  1.00  0.92  0.71  
R  0.71  0.92  1.00  −1.00j  −0.92j  −0.71j  −0.38j  −0.00j  0.00  0.38  0.71 
The orange bars show the encoding phase holes. It is unknown how they were implemented in practice.
DIR  F  RF  R  R  RB  B  LB  L  L  LF  F  

L  0.50  0.26  0.00  0.00  0.26  0.50  0.65  0.71  0.71  0.65  0.50  
Lj  0.50j  0.26j  0.00j  −0.00j  −0.26j  −0.50j  −0.65j  −0.71j  0.71j  0.65j  0.50j  
R  0.50  0.65  0.71  0.71  0.65  0.50  0.26  0.00  0.00  0.26  0.50  
Rj  −0.50j  −0.65j  −0.71j  0.71j  0.65j  0.50j  0.26j  0.00j  −0.00j  −0.26j  −0.50j 
Most matrix systems that are not great circle matrix systems also have at least one point where panning must be interrupted for a phase change as the sound is panned completely around the listener.
Example: Scheiber system and Sansui QS:
Value  Original system  ×  Resulting System 
Value  Original system  ×  Resulting System 


System  Scheiber  QS  System  QS  Scheiber  
Major Value  0.92  × 1.08 =  1.00  Major Value  1.00  × 0.92 =  0.92  
Minor Value  0.38  × 1.08 =  0.41  Minor Value  0.41  × 0.92 =  0.38  
Resultant Value  1.00  × 1.08 =  1.08  Resultant Value  1.08  × 0.92 =  1.00 
The procedure to find out what to multiply the original system coefficients by to get the resulting system is:
Percent Blend  Major is 1.00  Result is 1.00  Encode Angle  

Minor  Result  Major  Minor  Stylus θ  
100%  1.00  1.41  0.71  0.71  45.0° 
80%  0.80  1.28  0.78  0.62  38.7° 
50%  0.50  1.12  0.89  0.46  26.6° 
41%  0.41  1.08  0.92  0.38  22.5° 
32%  0.32  1.05  0.95  0.30  22.5° 
30%  0.30  1.04  0.96  0.29  16.7° 
25%  0.25  1.03  0.97  0.24  14.0° 
20%  0.20  1.02  0.98  0.20  11.3° 
Each row is the same matrix encoding, shown using different coefficients.
Input Values  ΔXEF (XE=1)  Angle  ΔXDO (XO=1)  Poincaré Sphere (r = 1)  Sphere Separation  Stylus Motion  

Matrix used with 
% Blend  dB  tan(θ) FE opp 
sec(θ) XF hyp 
xsc(θ) OF hyp 
csc(θ) XC hyp 
xsc(θ) BC hyp 
∠EXF ZXY θ 
∠ZOY 2θ 
cos(θ) XD adj 
sin(θ) OD opp 
∠XOZ α  XZ Chord 
OD ⊥ XZ Chord 
OM ⊥ YZ Chord 
OU ⊥ VW Chord 
OD dB  OM dB 
OU dB 
to opp chan ∠PXE 
to adj chan ∠AXE 
Calc:  bl %  vdB  = bl  sec(θ)  XF1  csc(θ)  XC1  atn(FE)  2θ  cos(θ)  sin(θ)  1802θ  2sin(α/2)  cos(α/2)  cos(θ)  sin(2θ)  vdB  vdB  vdB  90θ  θ 
SQb,HAb  100.0  0.0  1.000  1.414  0.414  1.414  0.414  45.0  90.0  0.707  0.707  90.0  1.414  0.707  0.707  1.000  3.01  3.01  0.00  45.0  45.0 
EV4bd  80.0  1.9  0.800  1.281  0.281  1.601  0.601  38.7  77.3  0.781  0.625  102.7  1.562  0.625  0.781  0.976  4.09  2.15  0.21  51.3  38.7 
Nar  67.0  3.5  0.670  1.204  0.204  1.797  0.797  33.8  67.6  0.831  0.557  112.4  1.662  0.557  0.831  0.925  5.09  1.61  0.68  56.2  33.8 
Nar2  60.0  4.4  0.600  1.166  0.166  1.944  0.944  31.0  61.9  0.857  0.514  118.1  1.715  0.514  0.857  0.882  5.77  1.34  1.09  59.0  31.0 
DQbd  57.7  4.8  0.577  1.155  0.155  2.001  1.001  30.0  60.0  0.866  0.500  120.0  1.732  0.500  0.866  0.866  6.03  1.25  1.25  60.0  30.0 
EV4be,DQbe  50.0  6.0  0.500  1.118  0.118  2.236  1.236  26.6  53.1  0.894  0.447  126.9  1.789  0.447  0.894  0.800  6.99  0.97  1.94  63.4  26.6 
QS,UM,H,HAf  41.4  7.7  0.414  1.082  0.082  2.614  1.614  22.5  45.0  0.924  0.383  135.0  1.848  0.383  0.924  0.707  8.35  0.69  3.01  67.5  22.5 
SQBbd,EVUbd  40.0  8.0  0.400  1.077  0.077  2.693  1.693  21.8  43.6  0.928  0.371  136.4  1.857  0.371  0.928  0.690  8.60  0.64  3.23  68.2  21.8 
UQE  31.8  10.0  0.318  1.049  0.049  3.300  2.300  17.6  35.3  0.953  0.303  144.7  1.906  0.303  0.953  0.578  10.4  0.42  4.77  72.4  17.6 
EV4fe  30.0  10.5  0.300  1.044  0.044  3.480  2.480  16.7  33.4  0.958  0.287  146.6  1.916  0.287  0.958  0.550  10.8  0.37  5.19  73.3  16.7 
DQfe  25.0  12.0  0.250  1.031  0.031  4.123  3.123  14.0  28.1  0.970  0.243  151.9  1.940  0.243  0.970  0.471  12.3  0.26  6.55  76.0  14.0 
EV4fd,EVUfd  20.0  14.0  0.200  1.020  0.020  5.099  4.099  11.3  22.6  0.981  0.196  157.4  1.961  0.196  0.981  0.385  14.2  0.17  8.30  78.7  11.3 
SQBfd  10.0  20.0  0.100  1.005  0.005  10.050  9.050  5.7  11.4  0.995  0.100  168.6  1.990  0.100  0.995  0.198  20.0  0.04  14.0  84.3  5.7 
Wide  5.0  26.0  0.050  1.001  0.001  20.025  19.025  2.9  5.7  0.999  0.050  174.3  1.998  0.050  0.999  0.100  26.0  0.01  20.0  87.1  2.9 
DQfd  0.0  80.0  0.000  1.000  0.000  99.000  98.001  0.0  0.0  1.000  0.000  180.0  2.000  0.000  1.000  0.000  80.0  0.00  80.0  90.0  0.0 
Matrix  Blend  dB  FE  XF  OF  XC  BC  θ  2θ  XD  OD  α  XZ  OD  OM  OU  OD dB  OM dB  OU dB  ∠PXE  ∠AXE 
Here are all of the items above put into one table.
The variables in the table are:
MATRIX COEFFICIENTS
USING ΔXEF
USING ΔCEX
USING ΔXAE and ΔXDO
TRIGONOMETRIC AND GEOMETRIC IDENTITIES
STYLUS VECTORS (Lower diagram for proper orientation to a record groove)
POINCARÉ SPHERE VALUES
SPHERE ANGLES
SPHERE ANGLE OPPOSITE
SPHERE ANGLE ADJACENT
SPHERE ANGLE CODED TO CODED
Hans Wallach, Edwin Newman, and Mark Rosenzweig defined the precedence effect in 1949. Helmut Haas later investigated the subject more in 1949. They showed that when a sound reaches the ears, followed by an identical sound from a different direction, the listener hears only one sound where the first sound came from.
The delay between the sounds can be as long as:
 5 ms for clicks
 40 ms for complex sounds (speech or music).
Larger delays are heard as echoes.
Human hearing uses these time and level differences in sounds reaching each ear to indicate the direction each sound comes from. A special microphone array is used to create a surround field. It encodes a quadraphonic matrix signal with the time delays already included to give additional clues to correctly locate the sound. No separation enhancement is needed in playback.
Here is a table of the relative levels and delays (in microseconds) received on the input microphones l, r, and s of such a recording for different directions of sound sources:
Angle  Level l  Delay l 
Level r  Delay r 
Level s  Delay s 
Direction  1st  2nd  3rd  
0  0.866  0  0.866  0  ~ 0.200  885  Straight Ahead  l r  s  
30  0.707  369  0.966  0  0.259  951  r  l  s  
60  0.500  639  1.000  0  0.500  762  Right Mic  r  l s  
90  0.259  737  0.966  0  0.707  369  Straight Right  r  s  l  
120  ~ 0.200  762  0.866  123  0.866  0  r s  l  
180  0.500  885  0.500  885  1.000  0  Straight Back Mic  s  l r  
240  0.866  123  ~ 0.200  762  0.866  0  l s  r  
270  0.966  0  0.259  737  0.707  369  Straight Left  l  s  r  
300  1.000  0  0.500  639  0.500  762  Left Mic  l  r s  
330  0.966  0  0.707  369  0.259  951  l  r  s  
360  0.866  0  0.866  0  ~ 0.200  885  Straight Ahead  l r  s 
~ Approximate value  Spill sound from room reflections into mics.
The stereo recording is matrixed from these three channels:
Decoding works with Dolby Surround or with any regular matrix (RM or QM) decoder.
Notice that this idea works only with a greatcircle matrix. It won't work with phase matrix systems.
The separations in a quadraphonic program that the listener notices the most are:
The other separations are not as noticeable. The separation between left back and right back is usually the least noticed.
Normal leveldifferential panning techniques cannot produce an image between these speakers unless the listener turns his head so that one each ear is facing each speaker. But there are several ways to make the side image appear, and some of them use normal panning techniques when the recording is mixed:
Any surround sound system with speakers located directly to the left and directly to the right of the listener does not have the sidelocation problem.
 The Dynaco Diamond has these speakers and unintentionally fixed this problem.
 The Denon QX Dual Triphonic system also unintentionally fixed this problem.
This method uses the precedence effect to make the separations seem larger. It introduced a phasereversed and delayed version of each decoded channel into the diagonally opposite channel.
 It did increase the apparent separation.
 It fixed the side location problem.
 It also caused an annoying tension feeling from phase reversed sound.
This uses microphones spaced the same distance apart that human ears are spaced. The microphones are aimed to the sides.
The listener listens to a recording from those microphones through earphones. The effect is very real, but there is no sound from the front or the back even if such sounds were there.
The effect is also audible with speakers. Quadraphonic speakers using an RM or QM matrix will place the images between the side speakers with the listener's head facing the front. This places images between the speakers, but does not produce a quadraphonic recording.
Dolby Surround plays a normally matrixed Dolby Surround, RM, or QM recording, decoding the dialog, left, right, and surround channels. The surround channels are delayed by the decoder to prevent surround leakage into the front from being noticed.
But this delay also uses the precedence effect to cause the side images to be located by the listener correctly as it was panned.
 It requires the listener to continue facing forward to hear all of the location effects correctly.
This method uses the precedence effect to make the separations seem larger. The difference is that it is in the recording. Nothing is added to the decoding equipment.
This is a surround (or Spheround) version of binaural recording, with delays in both leftright and frontback directions (and updown directions too for Spheround). So the position of the listener's head is unimportant.
 A special microphone technique creates a sound field. The human hearing system does the rest.
 This is like binaural recording, but works in a plane or even a solid space.
 It increases the apparent separation.
 No other separation enhancement is needed.
 It fixes the sidelocation problem.
 It gives a natural "you are there" effect that is a characteristic of sound fields.
The effect here is having so many channels that at least one or two pairs of channels are at opposite sides of the listener's head. This removes the sidelocation problem.
The Octophonic system works well.
There are several different methods:
In this method, the gain of each channel is quickly varied to make the dominant channels louder and the other channels quieter. This seemed to make the separation larger, but it had two disadvantages:
 It reduced the level of any ambience in the recording.
 It had the effect of "pumping" (where changes in signal levels or ambience caused by gain changes are noticed by the listener).
This method uses the precedence effect to make the separations seem larger. It introduced a phasereversed and delayed version of each decoded channel into the diagonally opposite channel.
 It did increase the apparent separation.
 It fixed the side location problem.
 It also caused an annoying tension feeling from phase reversed sound.
This method is used in the Sansui Variomatrix, the CBS Variblend, Dolby ProLogic, the ElectroVoice separation enhancement, and UniQuad Autovary.
 This looks for where the strongest signal is and momentarily adjusts the matrix to increase separation for that signal.
 The reduced separations for the other channels are not normally noticed.
 The upper diagram shows the normal separation pattern for Dolby Surround.
 The lower diagram shows the instantaneous changes Dolby Surround makes with a dominant frontcenter soloist.
 Decoding moves away from dominant sources. Front separation widens while back separation narrows with frontcenter solos.
DECODE EQUATIONS  DECODE RESULTS  Image  

Without ProLogic  
Left_Speaker  = 1.00 Left_Matrix  Left_Speaker  = 1.00 Left + .707 Dialog + .707 j Surround  
Right_Speaker  = 1.00 Right_Matrix  Right_Speaker  = 1.00 Right + .707 Dialog  .707 j Surround  
Dialog_Speaker  = .707 Left_Matrix + .707 Right_Matrix  Dialog_Speaker  = 1.00 Dialog + .707 Left + .707 Right  
Surround_Speaker  = .707 Right_Matrix  .707 Left_Matrix  Surround_Speaker  = 1.00 Surround + .707 Left  .707 Right  
With ProLogic (front solo)  
Left_Speaker  = .924 Left_Matrix  .383 Right_Matrix  Left_Speaker  = .924 Left + .383 Dialog + .924 j Surround  
Right_Speaker  = .924 Right_Matrix  .383 Left_Matrix  Right_Speaker  = .924 Right + .383 Dialog  .924 j Surround  
Dialog_Speaker  = .707 Left_Matrix + .707 Right_Matrix  Dialog_Speaker  = 1.00 Dialog + .707 Left + .707 Right  
Surround_Speaker  = .707 Right_Matrix  .707 Left_Matrix  Surround_Speaker  = 1.00 Surround  .707 Left  .707 Right 
Pumping is inaudible using this method because the signal levels are unchanged. Only the instantaneously perceived directions of the low level signals are changed. These are not normally noticed because human hearing normally notices the direction of the dominant signal over the other signals.
Dolby Surround plays a normally matrixed Dolby Surround, RM, or QM recording, decoding the dialog, left, right, and surround channels. The surround channels are delayed by the decoder.
This delay prevents surround leakage into the front from being noticed.
This delay prevents front leakage into the surround channels from being noticed.
This delay also uses the precedence effect to cause the side images to be located correctly as panned by the listener.
 It requires the listener to continue facing forward to hear all of the location effects correctly.
This method uses the precedence effect to make the separations seem larger. The difference is that it is in the recording. Nothing is added to the decoding equipment.
See "Theorem 7: Surround Fields" above and the Surround Fields page.
 A special microphone technique creates a sound field. The human hearing system does the rest.
 It increases the apparent separation.
 No other separation enhancement is needed.
 It fixes the sidelocation problem.
 It gives a natural "you are there" effect that is a characteristic of sound fields.
Each sound comes from a wall of sound, not a single point. This does the following:
 It makes it useful from more locations in the room.
 It creates a linear wave front instead of multiple spherical wave fronts.
 It keeps the ears from finding a single speaker in the room.
Thus, the ears locate the correct direction of the sound as panned.
This procedure was developed by the page author through empirical experimentation. But it seems to work very well:
The page author wanted to create a diagram of room size as shown in Leonard Feldman's book "Four Channel Sound" on page 62.
Value A  Separation between front channels 
Value B  Separation between back channels 
Value C  Separation between front and back channels on each side 
Value D  Separation between each front channel and the opposite stereo channel 
Value E  Separation between each back channel and the opposite stereo channel 
Value F  Separation between each back channel and a mono signal 
Value G  Separation between the center back signal and a mono signal 
Value H  Separation between each front channel and the center back channel 
Value K  Separation between each front channel and the opposite back channel 
Value S  Normal stereo separation of the record (assumed 40 dB) 
Value T  Separation between a stereo channel and mono (3 dB) 
Value U  Separation between a stereo channel and the center back signal 
For values of 0 dB, he had to use 0.05 dB to keep Excel from giving an error.
He took the common logarithm of each of the dB values above.
This gave a scale of values from 1.3 to 1.6 for values from .05 dB to 40 dB.
He then centered the values by subtracting 0.5 from each value for the final effect.
This gave a scale of values from 1.8 to 1.1 for values from .05 dB to 40 dB.
This would expand the space for large separations while contracting it for small ones. It also gave the wanted appearance.
Matrix  A  B  C  D  E  F  G  H  K  S  T  U 

QS  3  3  3  8.3  8.3  8.3  40  8.3  40  40  3  3 
0.02  0.02  0.02  0.42  0.42  0.42  1.10  0.42  1.10  1.10  0.02  0.02  
EV4  8.3  0.2  4.9  14  4.1  19  40  5.1  5.3  40  3  3 
0.42  1.2  0.19  0.65  0.11  0.78  1.10  0.21  0.22  1.10  0.02  0.02 
This frame is a 4unit by 4unit square centered on the origin of the graph.
This plot area is an 8unit by 8unit box centered on the origin of the graph.
W is the width at the front. Y is the width at the back.
X is the depth at the front. Z is the depth at the back.
LOCATION  DIRECTION  SAME SIDE  LET  OPP INPUT  LET  DIAGONAL  LET  MIRROR  OFFSET  FORMULA 

COEFFICIENTS:  1.0  0.1  0.01  +2  
Right Front  Horizontal  Left Front  A  Left Input  D  Left Back  K  Left Front  +2  'W = 1.0 A + 0.1 D + 0.01 K + 2 
Left Front  Horizontal  Right Front  A  Right Input  D  Right Back  K  Right Front  +2  'W = 1.0 A  0.1 D  0.01 K  2 
Right Front  Vertical  Right Back  C  Center Back  H  Left Back  K  Left Front  +2  'X = 1.0 C + 0.1 H + 0.01 K + 2 
Left Front  Vertical  Left Back  C  Center Back  H  Right Back  K  Right Front  +2  'X = 1.0 C + 0.1 H + 0.01 K + 2 
Right Back  Horizontal  Left Back  B  Left Input  E  Left Front  K  Left Back  +2  'Y = 1.0 B + 0.1 E + 0.01 K + 2 
Left Back  Horizontal  Right Back  B  Right Input  E  Right Front  K  Right Back  +2  'Y = 1.0 B  0.1 E  0.01 K  2 
Right Back  Vertical  Right Front  C  Mono  F  Left Front  K  Left Back  +2  'Z = 1.0 C  0.1 F  0.01 K  2 
Left Back  Vertical  Left Front  C  Mono  F  Right Front  K  Right Back  +2  'Z = 1.0 C  0.1 F  0.01 K  2 
A second set was needed for diamond speaker layouts:
LOCATION  DIRECTION  SAME SIDE  LET  OPP INPUT  LET  DIAGONAL  LET  MIRROR  OFFSET  FORMULA 

COEFFICIENTS:  1.0  0.1  0.1  +2  
Front  Horizontal  0  'W = 0  
Back  Horizontal  0  'W = 0  
Front  Vertical  Back  G  Left Input  T  Right Input  T  Back  +2  'X = 1.0 G + 0.1 T + 0.1 T + 2 
Back  Vertical  Front  G  Left Input  U  Right Input  U  Front  +2  'X = 1.0 G  0.1 U  0.1 U  2 
Right  Horizontal  Left  S  Mono  T  Back  U  Left  +2  'Y = 1.0 S + 0.1 T + 0.1 U + 2 
Left  Horizontal  Right  S  Mono  T  Back  U  Right  +2  'Y = 1.0 S  0.1 T  0.1 U  2 
Right  Vertical  0  'Z = 0  
Left  Vertical  0  'Z = 0 
Provide a table row for the x coordinates and a row for the y coordinates
The points are in the order lb, lf, rf, rb, lb (for diamond pattern: l, f, r, b, l).
The lb point is repeated to complete the quadrilateral (repeat l for the diamond pattern).
Use the plot option: Scatter with straight lines.
Select the orange values to make the QS plot.
EFFECT VALUES  VALUES  lb  lf  rf  rb  lb  IMAGE  

Matrix  A  B  C  D  E  F  G  H  K  S  T  U  W  Y  x  Y  W  +W  +Y  Y  
X  Z  y  Z  X  X  Z  Z  
QS  3  3  3  8.3  8.3  8.3  40  8.3  40  40  3  3  2.03  2.03  x  2.03  2.03  2.03 
2.03  2.03 

effect  0.02  0.02  0.02  0.42  0.42  0.42  1.10  0.42  1.10  1.10  0.02  0.02  2.03  2.03  y  2.03  2.03  2.03 
2.03  2.03 

EV4  8.3  0.2  4.9  14  4.1  19  40  5.1  5.3  40  3  3  2.49  0.81  x  0.81  2.49  2.49  0.81  0.81  
effect  0.42  1.2  0.19  0.65  0.11  0.78  1.10  0.21  0.22  1.10  0.02  0.02  2.21  2.27  y  2.27  2.21  2.21  2.27  2.27  
EFFECT VALUES  VALUES  l  f  r  b  l  IMAGE  
Matrix  A  B  C  D  E  F  G  H  K  S  T  U  W  Y  x  Y  W  +Y  +W  Y  
X  Z  y  Z  +X  +Z  X  Z  
DQ  3  3  3  8.3  8.3  8.3  40  8.3  40  40  3  3  0  3.10  x  3.10  0  3.10  0  3.10  
effect  0.02  0.02  0.02  0.42  0.42  0.42  1.10  0.42  1.10  1.10  0.02  0.02  3.10  0  y  0  3.10  0  3.10  0 